Source: ECON 300 E, University of Washington
Difficulty: Intermediate | Prerequisites: Consumer theory basics (utility maximisation, budget constraints), basic calculus (derivatives, product rule).
This set of topics extends consumer theory in three directions. The labour supply model treats the work-versus-leisure decision as a utility maximisation problem, connecting consumer choice to the labour market. Market demand aggregation shows how individual demand curves combine into a market-level curve. The revenue decomposition result links percentage changes in price and quantity to percentage changes in total revenue, a tool used throughout applied economics. All three topics draw on the same optimisation toolkit covered in the consumer theory unit.
A worker maximises utility over consumption and leisure, trading off hours of work against free time. Market demand is the horizontal sum of individual demand curves, with kinks where consumers drop out at high prices. Total revenue's percentage change equals the percentage change in price plus the percentage change in quantity, proven with the product rule.
Labour supply function
The number of hours a worker chooses to work as a function of the wage rate (and possibly non-wage income). In simple terms, it answers: "How many hours will you work if I pay you w per hour?"
Leisure
Time spent not working. In the labour model, it is treated as a good the consumer values, just like consumption. If there are 24 hours in a day and you work L, your leisure is h = 24 - L.
Non-wage income (unearned income)
Income received regardless of hours worked, such as an inheritance, dividends, or a trust payment. It shifts the budget constraint upward and typically reduces hours worked (a pure income effect).
Market demand curve
The total quantity demanded by all consumers at each price, obtained by summing individual demand curves horizontally (adding quantities at each price). Think of it as "everyone's demand stacked side by side."
Horizontal summation
The method of adding individual demand curves: at each price, add the quantities. This is distinct from vertical summation (used for public goods), where you add willingness to pay.
Total revenue / total expenditure
Price times quantity (P · Q). It is revenue from the firm's perspective and expenditure from the consumer's. The percentage change in P · Q equals the percentage change in P plus the percentage change in Q.
Roger's utility is U(c, h) = c · h, where c is daily consumption (dollars) and h is leisure hours.
With 24 hours in a day and L hours worked at wage w: c = wL and h = 24 - L.
Substituting: U = wL · (24 - L) = 24wL - wL^2.
Take the first-order condition: dU/dL = 24w - 2wL = 0.
Solve: L = 12. Roger works 12 hours regardless of the wage.
The wage cancels out because the utility function is multiplicative in c and h, so the substitution and income effects of a wage change exactly offset each other.
With U = c · h and no non-wage income, the optimal split is always half the day working, half at leisure. This is because the two "goods" (consumption and leisure) enter symmetrically when consumption equals wL.
Roger now receives $10 per day regardless of work. His consumption becomes c = wL + 10.
The new utility: U = (wL + 10)(24 - L) = 24wL - wL^2 + 240 - 10L.
dU/dL = 24w - 2wL - 10 = 0.
Solving: L = 12 - 5/w.
At w = 10: L = 12 - 0.5 = 11.5 hours.
Non-wage income makes Roger work less. This is a pure income effect: he is richer, so he "buys" more leisure.
The labour supply now depends on the wage: L = 12 - 5/w. Higher wages bring L closer to 12 (the non-wage income matters less). Lower wages push L further below 12.
If the wage is very low (w < 5/12), this formula would give L < 0, which does not make sense. In practice, Roger would choose not to work at all (a corner solution).
Rahul's demand: P = 30 - 2Qi, so Qi = 15 - 0.5P (active for P ≤ 30).
Juan's demand: P = 60 - 2Qj, so Qj = 30 - 0.5P (active for P ≤ 60).
You add quantities at each price, but only for consumers who are active at that price.
For 30 < P ≤ 60: only Juan is buying. Market Q = 30 - 0.5P, or equivalently P = 60 - 2Q.
For 0 < P ≤ 30: both are buying. Market Q = (15 - 0.5P) + (30 - 0.5P) = 45 - P, or equivalently P = 45 - Q.
The market demand curve has a kink at P = 30 (Q = 15). Above that price, the slope is -2 (only Juan). Below it, the slope is -1 (both consumers). Always check where individual curves hit zero before summing.
%ΔR ≈ %ΔP + %ΔQ, where R = P · Q.
R = P · Q. Differentiate with respect to some variable t: dR/dt = P · (dQ/dt) + Q · (dP/dt).
Divide both sides by R = P · Q: (dR/dt) / (P·Q) = (dQ/dt)/Q + (dP/dt)/P.
The left side is the percentage change in R. The right side is the percentage change in Q plus the percentage change in P.
This is a direct application of the product rule from calculus. The result is exact for infinitesimal changes and a close approximation for small discrete changes. It breaks down for large changes (e.g., if price doubles and quantity doubles, revenue quadruples, not merely doubles plus doubles).
Labour supply (U = ch, no non-wage income): L = 12 (constant, independent of wage).
Labour supply (U = ch, with non-wage income m): L = 12 - m/(2w).
Individual demand (Rahul): Qi = 15 - 0.5P (for P ≤ 30).
Individual demand (Juan): Qj = 30 - 0.5P (for P ≤ 60).
Market demand: Q = 30 - 0.5P for 30 < P ≤ 60; Q = 45 - P for 0 < P ≤ 30.
Revenue decomposition: %ΔR = %ΔP + %ΔQ (for small changes).
Students sometimes add demand curves vertically (adding prices at each quantity) instead of horizontally (adding quantities at each price). Vertical summation is used for public goods, not private goods.
In the labour model, students forget that the wage rate is the "price" of leisure, not the price of consumption. Leisure costs you w per hour in foregone earnings.
The revenue decomposition %ΔR = %ΔP + %ΔQ is an approximation for small changes. Students sometimes apply it to large percentage changes and get confused when the numbers do not add up exactly.
When non-wage income enters, the labour supply depends on the wage. Students sometimes leave the answer as L = 12 without re-deriving the FOC with the new budget constraint.
⚠️ The labour supply setup (substituting c = wL and h = 24 - L into U = ch) is a common exam problem. Know how to set it up cleanly.
⚠️ When non-wage income is added, be ready to re-derive the FOC from scratch. Do not try to adjust the old answer.
⚠️ For market demand, always identify the price at which each consumer drops out before summing. The kink is where marks are earned or lost.
⚠️ The revenue decomposition proof is a "show that" question, meaning you must write out the calculus step by step. Stating the result is not enough.
True or false: If U = ch with no non-wage income, doubling the wage doubles the hours worked.
Fill in the blank: Market demand is obtained by ______ summation of individual demand curves.
True or false: Non-wage income causes a pure substitution effect on labour supply.
Fill in the blank: %ΔR ≈ %ΔP + %ΔQ relies on the ______ from calculus.
True or false: A market demand curve can have kinks even though each individual demand curve is a straight line.
Q: Given U(c,h) = ch with c = wL and h = 24 - L, what is Roger's optimal L?
A: L = 12. The FOC gives 24w - 2wL = 0, and w cancels.
Q: If Roger receives $10/day in non-wage income, what is his new labour supply function?
A: L = 12 - 5/w. At w = 10, he works 11.5 hours.
Q: Why does non-wage income reduce hours worked?
A: It is a pure income effect. Roger is richer at any given wage, so he consumes more leisure (works less). There is no substitution effect because the wage (the price of leisure) has not changed.
Q: What is the market demand for root beer if Rahul has P = 30 - 2Qi and Juan has P = 60 - 2Qj?
A: For 30 < P ≤ 60, only Juan buys: Q = 30 - 0.5P. For 0 < P ≤ 30, both buy: Q = 45 - P. The curve has a kink at P = 30.
Q: Using calculus, show that %ΔR = %ΔP + %ΔQ.
A: R = PQ. By the product rule: dR = PdQ + QdP. Divide by R = PQ: dR/(PQ) = dQ/Q + dP/P, which is %ΔR = %ΔQ + %ΔP.
The labour supply model connects to the Slutsky decomposition: a wage increase has both a substitution effect (leisure becomes more expensive, so work more) and an income effect (richer, so consume more leisure). With U = ch and no non-wage income, these exactly cancel. Market demand aggregation reappears in welfare economics and surplus analysis. The revenue decomposition is the basis for understanding elasticity: if %ΔQ > %ΔP in magnitude, demand is elastic and a price increase reduces revenue.
Labour supply, leisure-consumption tradeoff, work-leisure model, non-wage income, unearned income, inheritance effect on labour, backward-bending labour supply, market demand curve, horizontal summation, individual demand, aggregate demand, kink in demand curve, total revenue, total expenditure, percentage change, product rule, revenue decomposition, elasticity, ECON 300, intermediate microeconomics.